Problem 0: Complete the Canvas quiz "PSet 2 - Canvas".
Problem (10 points):
Determine if the following argument form is valid or invalid .
Show your work.
$p \vee q \vee r$
$\neg p$
$q \rightarrow r$
$\therefore r$
Problem (8 points): Show how to come to the conclusion $v$ given the following premises. Show which rule of inference you used at each step and which premises and/or previous conclusions you applied them to.
$p \vee \neg q \vee r$
$p \rightarrow s$
$s \rightarrow v$
$\neg q \rightarrow \neg u$
$u$
$r \rightarrow v$
Problem (8 points): You meet four people on the Island of Knights and Knaves. Given the following statements they make, determine for each individual whether they are a knight or a knave. Show your reasoning. (Consider each utterance as a single (possibly compound) statement. Knights' statements are always true and knaves' statements are always false. )
- A: I am a knight and C is a knave
- B: A is a knave
- C: there is only one knight and it is not me
- D: A and myself are knights and B and C are knaves.
Problem (10 points): Let $P$ be the set of players, $T$ be the set of professional teams, and $C$ be the set of college teams. Let $M(x,y)$ be the predicate "player $x$ plays for pro team $y$" and $N(x,y)$ be the predicate "player $x$ played for college team $y$".
Write each of the following in predicate logic. Be sure to indicate the domain of each quantified variable using $P$, $T$, or $C$. Assume that the specific nouns mentioned are members of the appropriate sets.
- No player played for Harvard. [note: Harvard is a college team]
- Every player who plays for Golden State played for Virginia. [note: Golden State is a pro team and Virginia is a college team]
- Some player didn't play for any college team.
- For every college team, there is a player who played for them and plays for a pro team.
- Every player played for a unique college team. [note: use the standard quantifiers $\forall$ and $\exists$; you may also use $=$ (which is really a two-place predicate written using infix notation rather than function notation, so $x = y$ can be thought of as the predicate $=(x, y)$ meaning "x and y are the same element")]